Deform a quadrant of a cylinder along the z-axis
If you want a very smooth quadrant you will need more vertex density.  You know a smooth human face with many contours requires many more vertices than a simple cube
Edit Mode
Loop cut can do this.Control R
You can set the cursor to the center of a completed cylinder .. as if you had 4 quadrants.  You probably have seen Cylinder and Cones being edited which have complete circles so Blender can figure out center easily.  You need this one extra step since you only have one quarter circle.
Set cursor . Shift S .Set the center of scale for proportional editing to cursor.  This allows you to change shape from any point you choose.  This item is two UI boxes to the left of [edit mode] on the bottom strip of 3D View Window in your images above
Select bottom of quadrant. Many vertices.
Enable Proportional edit. o. Scale to see effect. Use mouse wheel.
Experiment many more times
State whether or not you are making progress
Tutorial https://www.youtube.com/watch?v=MrNVVHdKd3E